Global dynamics of a Yang-Mills field on an asymptotically hyperbolic space

TL;DR

This paper investigates the behavior of a spherically symmetric SU(2) Yang-Mills field on an asymptotically hyperbolic space, revealing bifurcation patterns, static solutions, and relaxation dynamics using a hyperboloidal approach.

Contribution

It uncovers the bifurcation structure of static solutions and analyzes the relaxation dynamics of the Yang-Mills field on a hyperbolic spacetime.

Findings

Bifurcation pattern of static solutions related to Morse index

Relaxation to ground state for generic initial data

Unstable solutions for specific initial conditions

Abstract

We consider a spherically symmetric (purely magnetic) SU(2) Yang-Mills field propagating on an ultrastatic spacetime with two asymptotically hyperbolic regions connected by a throat of radius $\alpha$. Static solutions in this model are shown to exhibit an interesting bifurcation pattern in the parameter $\alpha$. We relate this pattern to the Morse index of the static solution with maximal energy. Using a hyperboloidal approach to the initial value problem, we describe the relaxation to the ground state solution for generic initial data and unstable static solutions for initial data of codimension one, two, and three.